Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 65 / 3 / c / Solution

The proposed mixing length can be motivated as a local phenomenological estimate: on either half of a self-similar jet, the centreline supplies a distinguished reference plane, and an eddy at distance $|y|$ might retain momentum over a fixed fraction $C_1|y|$ of that transverse distance. This givesIt is compatible with the dilation $x\mapsto ax$, $y\mapsto ay$ and introduces no independent outer length. In the turbulent plane jet similarity regime its typical value at $|y|\sim \delta$ is $C_1\delta$, so the scaling from part (b) has $\delta /L\sim C_1^2$.

This argument is a modeling assumption, not a physical necessity: a jet centreline is not a wall that prevents turbulent eddies from crossing it. Thus a mixing length vanishing there cannot automatically be justified throughout the core. Parts (e) and (f) exhibit the precise difficulty: the resulting finite-velocity similarity profile has singular centreline shear and zero shear at a finite free edge, contradicting the printed edge definition. The local dimensional motivation does not make all the stipulated global jet properties consistent.